where equiv is an equivalence relation and fn is a function
symbol other than if, hide, force or case-split. Such
rules allow ``alternative'' definitions of fn to be proved as
theorems but used as definitions. These rules are not true
``definitions'' in the sense that they (a) cannot introduce new
function symbols and (b) do not have to be terminating recursion
schemes. They are just conditional rewrite rules that are
controlled the same way we control recursive definitions. We call
these ``definition rules'' or ``generalized definitions''.

Consider the general form above. Generalized definitions are stored
among the :rewrite rules for the function ``defined,'' fn above, but
the procedure for applying them is a little different. During
rewriting, instances of (fn a1 ... an) are replaced by corresponding
instances of body provided the hyps can be established as for a
:rewrite rule and the result of rewriting body satisfies the
criteria for function expansion. There are two primary criteria,
either of which permits expansion. The first is that the
``recursive'' calls of fn in the rewritten body have arguments that
already occur in the goal conjecture. The second is that the
``controlling'' arguments to fn are simpler in the rewritten body.

The notions of ``recursive call'' and ``controllers'' are
complicated by the provisions for mutually recursive definitions.
Consider a ``clique'' of mutually recursive definitions. Then a
``recursive call'' is a call to any function defined in the clique
and an argument is a ``controller'' if it is involved in the measure
that decreases in all recursive calls. These notions are precisely
defined by the definitional principle and do not necessarily make
sense in the context of generalized definitional equations as
implemented here.

But because the heuristics governing the use of generalized
definitions require these notions, it is generally up to the user to
specify which calls in body are to be considered recursive and what
the controlling arguments are. This information is specified in the
:clique and :controller-alist fields of the :definition rule class.

The :clique field is the list of function symbols to be considered
recursive calls of fn. In the case of a non-recursive definition,
the :clique field is empty; in a singly recursive definition, it
should consist of the singleton list containing fn; otherwise it
should be a list of all of the functions in the mutually recursive
clique with this definition of fn.

If the :clique field is not provided it defaults to nil if fn does
not occur as a function symbol in body and it defaults to the
singleton list containing fn otherwise. Thus, :clique must be
supplied by the user only when the generalized definition rule is to
be treated as one of several in a mutually recursive clique.

The :controller-alist is an alist that maps each function symbol in
the :clique to a mask specifying which arguments are considered
controllers. The mask for a given member of the clique, fn, must be
a list of t's and nil's of length equal to the arity of fn. A t
should be in each argument position that is considered a
``controller'' of the recursion. For a function admitted under the
principle of definition, an argument controls the recursion if it is
one of the arguments measured in the termination argument for the
function. But in generalized definition rules, the user is free to
designate any subset of the arguments as controllers. Failure to
choose wisely may result in the ``infinite expansion'' of
definitional rules but cannot render ACL2 unsound since the rule
being misused is a theorem.

If the :controller-alist is omitted it can sometimes be defaulted
automatically by the system. If the :clique is nil, the
:controller-alist defaults to nil. If the :clique is a singleton
containing fn, the :controller-alist defaults to the controller
alist computed by (defun fn args body). If the :clique contains
more than one function, the user must supply the :controller-alist
specifying the controllers for each function in the clique. This is
necessary since the system cannot determine and thus cannot analyze
the other definitional equations to be included in the clique.

For example, suppose fn1 and fn2 have been defined one way and it is
desired to make ``alternative'' mutually recursive definitions
available to the rewriter. Then one would prove two theorems and
store each as a :definition rule. These two theorems would exhibit
equations ``defining'' fn1 and fn2 in terms of each other. No
provision is here made for exhibiting these two equations as a
system of equations. One is proved and then the other. It just so
happens that the user intends them to be treated as mutually
recursive definitions. To achieve this end, both :definition rules
should specify the :clique(fn1 fn2) and should specify a suitable
:controller-alist. If, for example, the new definition of fn1 is
controlled by its first argument and the new definition of fn2 is
controlled by its second and third (and they each take three
arguments) then a suitable :controller-alist would be
((fn1 t nil nil) (fn2 nil t t)). The order of the pairs in the
alist is unimportant, but there must be a pair for each function in
the clique.

Note that the actual definition of fn1 has the runic name
(:definition fn1). The runic name of the alternative definition is
(:definition lemma), where lemma is the name given to the event that
created the generalized :definition rule. This allows theories to
switch between various ``definitions'' of the functions.

By default, a :definition rule establishes the so-called ``body'' of a
function. The body is used by :expandhints, and it is also used
heuristically by the theorem prover's preprocessing (the initial
simplification using ``simple'' rules that is controlled by the
preprocess symbol in :do-nothints), induction analysis, and the
determination for when to warn about non-recursive functions in rules. The
body is also used by some heuristics involving whether a function is
recursively defined, and by the expand, x, and x-dumb commands of
the proof-checker.

See rule-classes for a discussion of the optional field :install-body of
:definition rules, which controls whether a :definition rule is used
as described in the paragraph above. Note that even if :install-body nil
is supplied, the rewriter will still rewrite with the :definition rule;
in that case, ACL2 just won't install a new body for the top function symbol
of the left-hand side of the rule, which for example affects the application
:expand hints as described in the preceding paragraph. Also
see set-body and see show-bodies for how to change the body of a function
symbol.

Note only that if you prove a definition rule for function foo, say,
foo-new-def, you will need to refer to that definition as foo-new-def
or as (:DEFINITION foo-new-def). That is because a :definition rule
does not change the meaning of the symbol foo for :usehints,
nor does it change the meaning of the symbol foo in theory expressions;
see theories, in particular the discussion there of runic designators.
Similarly :pefoo and :pffoo will still show the
original definition of foo.

The definitional principle, defun, actually adds :definition
rules. Thus the handling of generalized definitions is exactly the
same as for ``real'' definitions because no distinction is made in
the implementation. Suppose (fn x y) is defun'd to be
body. Note that defun (or defuns or mutual-recursion)
can compute the clique for fn from the syntactic presentation and
it can compute the controllers from the termination analysis.
Provided the definition is admissible, defun adds the
:definition rule (equal (fn x y) body).